Question: Prison overcrowding the 1980s saw a trend toward oldfashioned punitive deterrence as opposed to the more liberal penal policies and community-based corrections popular in the 1960s and early 1970s. As a result, prisons became more crowded, and the gap between the number of people in prison and the prison capacity widened. The number of prisoners (in thousands) in federal and state prisons is approximated by the function
N(t) = 3.5t2 + 26.7t + 436.2 (0 ≤ t ≤ 10)
Where t is measured in years, with t = 0 corresponding to 1984. The number of inmates for which prisons were designed is given by
C(t) = 24.3t + 365 (0 ≤ t ≤ 10)
Where c(t) is measured in thousands and t has the same meaning as before. Show that the gap between the number of prisoners and the number for which the prisons were designed has been widening at any time t.
Hint: first, write a function g that gives the gap between the number of prisoners and the number for which the prisons were designed at any time t. Then show that g'(t) > 0 for all values of t in the interval (0, 10).